University of Arizona ECE/OPTI 500C: Photonic Communications Engineering I C Fall Forward Error Correction (FEC) By Ivan B.

Transcription

1 Uiversity of Arizoa ECE/OPTI 500C: Photoi Commuiatios Egieerig I C Fall 00 Forward Error Corretio (FEC) By Iva B. Djordjevi j

2 Codig for Optial Chaels Importat lasses of odes: Liear lok odes Cyli odes Covolutioal odes Turo odes Low-desity parity hek odes Liear odes Blok odes Cyli Noyli Covolutioal odes Noliear odes Blok odes Trellis odes I (,k) lok ode the hael eoder aepts iformatio i suessive k-it loks, for eah loks it adds -k redudat its that are algeraially related to the k message its; therey produig a overall eoded lok of its (>k), kow as a ode word) I ovolutioal l ode the eodig operatio may e osidered d as the disrete-time ti ovolutio of the iput sequee with the impulse respose of the eoder.

3 Liear Blok Codes Liearity property: A ode is said to e liear if ay two ode words i the ode a e added i modulo- arithmeti to produe a third ode word. Example: (,) repetitio ode. The repetitio ode has two ode words x 0 =(00 0) ad x =( ). The liear omiatio of two ode words is a ode word: x0 x0 x0 x x x x = x x x x The set of ode words from a liear lok ode forms a group uder the additio operatio, eause all-zero ode word serves as the idetity elemet, ad the ode word itself serves as the iverse elemet. This is the reaso why the liear lok odes are also alled the group odes. The liear lok ode (,k) a e oserved as a k-dimesioal suspae of the vetor spae of all -tuples over the iary filed GF(). All -tuples p over GF() form the vetor spae: -The sum of two -tuples a=(a a a ) ad =( ) is learly a -tuple =a+ = (a + a + a + )=( +a +a +a )=+a. The all-zero vetor 0=(0 0 0) is the idetity elemet, ad -tuple a itself is the iverse elemet a+a=0. Therefore, the -tuples form the Aelia group with respet to the additio operatio. 3

4 The salar multipliatio is defied y: a=( a a a ), GF() Distriutive laws: (a+)=a+ (+)a= a + a,, GF() Assoiate law: ( )a= (a) ( ) Clearly, the set of all -tuples is a vetor spae over GF(). The set of all ode words from a (,k) liear lok ode forms a aelia group uder the additio operatio. It a e show, i a fashio similar il to that t aove, that t all ode words of a (,k) liear lok odes form the vetor spae of dimesioality k. There exists k asis vetors (ode words) suh that every ode word is a liear omiatio of these ode words. Example: (,) repetitio ode: C={(0 0 0), ( )}. Two ode words i C a e represeted as liear omiatio of all-oes asis vetor: ( )= ( ), (00 0)= ( )+ ( ) Example: (6,3) ode: C={(000000), (000),(000),(00),(00),(000),(000),(00)}. It a e easily show that C is a 3-dimesioal spae with asis vetors g 0 =(00), g =(000), g =(000). 4

5 Geerator Matrix for Liear Blok Code Ay ode word x from the (,k) liear lok ode a e represeted as a liear omiatio of k asis vetors g i, i=0,,..,k-: g 0 m m... m g x g g g m, m = m m... m... g k 0 0 k k 0 k- g 0, = g x mg G Geerator matrix... g k k-it message word Example: Geerator matries for repetitio (,) ode, ad (,-) sigle-parityhek ode: G rep par G 5

6 Struture of systemati ode word m 0 m m m k- 0 -k- Message its Parity its Code word its : mi, i 0,,..., k xi i k, i k, k,..., Parity its : p m p m... p m i 0i 0 i k, i k p ij, if i depeds o 0, otherwise Matrix otatio : j u m mm... m... x xx... x 0 k 0 k 0 Coeffiiet matrix : p00 p0... p 0, k p0 p... p, k P pk,0 pk,... pk, k x m m I k P I idetity k matrix G Ik P geerator matrix x mg 6

7 Parity-Chek Matrix for Liear Blok Code Aother useful matrix assoiated with the liear lok odes is the parity-hek matrix. Let us expad the matrix equatio: x mg x x m 0 0 m k... xk mk x m p m p... m p k k k,0 xk m0p0m p... mk pk,... x m p m p... m p 0 0, k, k k k, k 7

8 The last -k equatios a e rewritte as: x p x p... x p x k k,0 k x p x p... x p x k k,0 k x p x p... x p x 0 0 0, k, k k k, k x x... x 0 T T xh 0, H P I k GH T 0 p00 p0... pk, p0 p... pk, p0, k p, k... pk, k k x P 0 T GH Ik P P P Ik T 8

9 Example: Parity-Chek Matries for (,) repetitio ode ad (,-) sigle-parity hek ode H rep H par... Dual Code Every(,k) liear lok ode with geerator matrix G ad parity-hek matrix H has a dual ode with geerator matrix H ad parity hek matrix G. Example: (,) repetitio ode ad (,-) sigle-parity hek ode are dual. 9

10 Sydrome: Defiitio ad Properties Reeived vetor : r xe e error vetor (patter) if a error oured i the i th loatio ei 0 otherwise Sydrome : s rh T Properties:. The sydrome depeds oly o the error patter, ad ot o the trasmitted ode word. s=(x+e)h( + T =xhh T +ehh T =ehh T 0

11 . All error patters that differ y a ode word have the same sydrome. For k message its there are k distit vetors deoted y x i (i=0,,, k- ). For ay error patter we a defied k distit vetors e i =e+x i, i=0,,, k - The oset of the ode: {e i=0 k i, 0,,, -} Sie e i H T =eh T +x i H T =eh T Eah oset of the ode, havig k elemets that differ at most y a ode vetor, is haraterized y a uique sydrome. (-k) elemets of the sydrome are liear omiatios of the elemets of the error patter: s 0 =e 0 +e -k p 00 +e -k- p 0 + +e - p k-,0 s =e +e -k p 0 +e -k- p + +e - p k-, s -k- =e -k- +e -k p 0,-k- +e -k- p,-k- + +e - p k-,-k- The system of equatios is uderdetermied (more ukows tha equatios) The kowledge of the sydrome redues the searh for the error patter from to -k possiilities.

12 3. The sydrome is the sum of those olums of the parity hek matrix orrespodig to the error loatios. T s eh h T T h T... e e... e ei i... i T h H h h h s h 4. With sydrome deodig, a (,k) liear lok ode a orret up to t errors per ode word, providig that the Hammig oud is satisfied: k t i0 i The total umer of sydromes, iludig the all-zero sydrome, is -k, ad eah sydrome orrespods to a speifi error patter. For a -it ode word there are hooses i multiple-error patters, where i is the umer of error loatios i error patter e. The total umer of all possile orretale error patters: t i0 i

13 The total umer of sydromes aot e less tha total umer of possile error patters. The iary ode for whih Hammig oud is satisfied with equality sig is alled the perfet ode. Miimum Distae Cosideratios Hammig distae etwee two ode words x ad x, d(x,x ), is defied as the umer of loatios i whih their respetive elemets differ. Hammig weight, w(x), of a ode vetor x is defied as the umer of ozero elemets i the vetors. The miimum distae, d mi, of a liear lok ode is defied as the smallest Hammig distae etwee ay pair of ode vetors i the ode. The miimum distae of a liear lok ode is the smallest Hammig weight of the ozero ode vetors i the ode. xh T =0 H=[h h h ], h i ith olum i H-matrix The miimum distae of a liear lok ode is defied y the miimum umer of olums of the H-matrix whose sum is equal to the zero vetor. 3

14 x i t r x ij t r x x i t r t x j (a) () (a) Hammig distae d(x i, x j ) t. () Hammig distae d(x i, x j ) t. The reeived vetor is deoted y r. A (,k) liear lok ode has the power to orret all error patters of weight t or less, if, ad oly if, d(x i,xx j )t+ for all x i ad x j. A (,k) liear lok ode of miimum distae d mi a orret up to t errors if, ad oly if, t /(d mi -) (*) where deotes the largest iteger less tha or equal to the elosed quatity. 4

15 Sydrome Deodig ad Stadard Array The k ode words partitio the spae of all reeived words ito k disjoit susets. Ay reeived word withi suset is deoded as the uique ode word. A stadard array is a tehique y whih this partitio a e ahieved. Stadard Array Costrutio:. Step. Write dow k ode words as elemets of the first row, with the all-zero odeword as the leadig elemet.. Step. Repeat the steps (a) ad () util all words are exhausted. (a) Out of the remaiig uused -tuples, selet oe with the least weight for the leadig elemet of the ext row (the urret row). () Complete the urret row y addig the leadig elemet to eah ozero ode word appearig i the first row ad writig dow the resultig sum i the orrespodig olum. 5

16 Stadard array for a (, k) lok ode: Coset leaders x 0 x x3... xi... x k e x e x e... xi e... x k e e x e x e... xi e... x k e e x e x e... x e... x e j j 3 j i j k j e x e x e... x e... x e 3 i k k k k k k Example: Stadard array of (6,3) ode: C={(000000), (000),(000),(00),(00),(000),(000),(00)}. Code words

17 Stadard array properties:. All -tuples of row are distit.. Eah -tuple t l appears exatly oe i the stadard d array. 3. There are exatly -k rows i the stadard array. 4. For perfet odes (satisfyig the Hammig oud with equality sig) all -tuples of weight t=it[(di t[(d mi -)/] or less appear as oset leaders (it[x] ]is the iteger part of x). ) 5. For quasi-perfet odes, i additio to all -tuples of weight t or less, some ut ot all - tuples of weight t+ appear as osset leaders. 6. All elemets i the same row (oset) have the same sydrome. 7. Elemets i differet rows have differet sydromes. 8. There are -k differet sydromes orrespodig to -k rows. The key to the deodig usig a stadard array is to view the osset leaders as error patters aused y BSC. Let e i represet the oset leader of ith row, ay elemet i that row a e represeted as y=x+e i, where x is the ode word lyig at the top of olum i whih h y lies, ad it is the losest ode word to y: d yx, w y x w e x x w e yx, y x e x x e x e d, d, i ei x H y x ei H y x ei x H i i d w w w H i i w w d w d w 7

18 The proaility of word error: i i P e w p p w i0 i Crossover proaility of BSC The umer of oset leaders of weight i The weight distriutio of the oset leaders: w i i,, i=0,,,,,, Example: The weight distriutio of oset leaders i (6,3) ode are w 0 =, w =6, w =, w i =0, i=3,,6; leadig to the word error proaility: w P e p p p p p P e 4 p p.40 w 5 p0 Beause all sigle errors ad oe doule error are orretale, the remaiig 4 doule errors domiate for small p 8

19 Deodig proedure (sydrome deodig): For the reeived vetor y, ompute the sydrome s=yh T. Notie that oe-to-oe orrespodee a e estalished etwee the sydromes ad error patters (Property 7), leadig to the lookup tale otaiig the error patter. Withi the oset haraterized y the sydrom s, idetify the oset leader (i.e., the error patter with the largest proaility of ouree); e 0. Compute the ode vetor x=y+e 0 as the deodig versio of the reeived vetor y. Example: y=(000) ( ) s=yh T =() ( ) e 0 =(00000) x=y+e 0 =(00) Sydrome (the address to the lookup tale) Error Patter The lookup tale for (6,3) ode

20 Hammig Codes A family of (,k) liear lok odes with followig parameters: Blok legth: = m - Numer of message its: k= m -m- Numer of parity its: -k=m where m3 are kow as Hammig odes. Example: (7,4) Hammig ode G H d mi 3 sie Hammig odes are sigle-error orretig iary perfet odes. 0

21 (7,4) Hammig ode words: Deodig tale for (7,4) Hammig ode: Message Code Word Weight of Word Code Word Sydrome Error Patter Code word [0000] is set, ad the reeived vetor [0000] is with error i the third it s [ 0000 ] 0 00 e

22 Codig Gai Codig gai refers to the savigs attaiale i the eergy per iformatio it to oise spetral desity ratio (E /N 0 ) required to ahieve a give it error proaility whe odig is used ompared to that with o odig. : BPSK re E k E ke E Bit error proaility : w e P t P 0 0 N re Q N E Q p, t w x p t P ratio : oise to sigal At high t N t w p p t N e P exp Sie t re x x Q t p t N t 0 exp N re P

23 Uoded it error proaility : Example: Codig Gai for (7,4) Hammig Code E P, exp u N P 7 w e p 7 p p i p p7i Codig gai : i i E / N0 G u h E / N0 Hard deisio : G h [db] 0log Soft deisio : r t 0 rt G s [db] 0log 0 rd mi p 3 P Pw e 7 give a ertai Pw p e 7 p 3 / 3 Q Required E / N0 BER re N 0 6 to ahieve the give BER 3

24 Cyli Codes. Liearity property: The sum of ay two ode words i the ode is also a ode word. C li t A li hift f d d i th d i l d d Cyli property: Ay yli shift of a ode word i the ode is also a ode word polyomial: word Code i i polyomial word a ode also mod i i Geerator Polyomial The polyomial + ad its fators play a major role i the geeratio of yli odes. Geerator polyomial k k i i i g g g() is a fator of + deg(g())= k g() is the least degree polyomial i the ode 4 g() is a fator of +, deg(g())=-k, g() is the least degree polyomial i the ode

25 A yli ode is uiquely determied y the geerator polyomial: l g a polyomia word a ode, m m m m ode yli Systemati k k 0... message polyomial: - g a g m k k k 0 remider the, Eoder for a (, k) yli ode. m C k k k 0... Eoder for a (, k) yli ode. 5

26 Sydrome Calulatio polyomial: word Reeived s s q r r r r r remider the... polyomial: word Reeived 0 polyomial sydrome the s k s s g q g deg remider the, Sydrome alulator for (, k) yli ode. 6

27 Properties of the sydrome polyomial:. The sydrome of reeived word polyomial is also the sydrome of orrespodig error polyomial.. Let s() e the sydrome of reeived word polyomial r(). The, the sydrome of r(), a yli shift of r(), is s(). 3. The sydrome polyomial s() is idetial to the error polyomial e(), assumig that the errors are ofied to the (-k) parity-hek its of the reeived word polyomial r(). Parity-Chek Polyomial h k i i k g i The geerator polyomial g() ad the parity-hek polyomial h() are fators of the +: g h mod 0 Geerator ad Parity-Chek Matries -tuples pertaiig to the k polyomials g(), g(),, k- g() may e used i rows of of the k-y- geerator matrix G. -tuples pertaiig to the (-k) polyomials k h( - ), k+ h( - ),, - h( - ) may e used i rows of the (-k)-y- parity-hek matrix H. 7

28 Hammig Codes Revisited [(7,4) Cyli Code Example] 7 +=(+)(+ + 3 )( 3 ) g() 3 h()=(+)(+ + 3 )= Message sequee: 00 m()= m C k -k m()= 3 m()= Eoder for the (7, 4) yli ode geerated y g() 3. Sydrome alulator for the (7 4) yli ode geerated y the polyomial g() 8 Sydrome alulator for the (7, 4) yli ode geerated y the polyomial g() 3.

29 g g g h h ' G g g ' H h h G H G E l M i l L th C d Example: Maximal-Legth Codes Maximal-legth odes are dual of Hammig odes: Blok legth: = m (m3) h g Blok legth: = m - (m3) Numer of message its: k=m Miimum distae: d mi = m- 3 ode : legth maximal (7,3) deg h m h 9 4 g h

30 Eoder for the (7, 3) maximal-legth legth ode; the iitial state of the eoder is show i the figure. Other Cyli Codes Cyli Reduday Chek Codes (CRC Codes) - Extremely well suited for error detetio - Error urst of legth B: a otiguous sequee of B its i whih the first ad ad the last its or ay other itermediate its are reeived i error. - (,k) CRC odes are apale of detetig:. All error ursts of legth -k.. A fratio of error ursts of legth equal to -k+; the fratio equals - -(-k-). 3. A fratio of error of legth greater tha -k+; the fratio equals - -(-k-). 4. All omiatios of d mi - (or fewer) errors. 5. All error patters with a odd umer of errors if the geerator polyomial g() for the ode has a eve umer of ozero oeffiiets. 30

31 CRC Codes Geerator polyomial -k CRC- ode CRC-6 ode (USA) CRC-ITU Bose-Chaudhuri-Hoqueghem (BCH) Codes Primitive BCH odes (t-error orretig odes): Blok legth: = m - (m3) Numer of message its: k-mt Miimum distae: d mi t+ Biary BCH odes of legth up to 5 -: k t Geerator Polyomial

32 0000 g Reed-Solomo Codes (RS Codes) RS odes are a importat sulass of oiary BCH odes A t-error orretig RS ode parameters: Blok legth: = m - symols Message size: k symols Parity-hek size: -k=t symols Miimum distae: d mi =t+ symols Coateated Codes The oateated ode (proposed y Forey), is a (N,KkDd) ode with the miimum distae of at least Dd: k -ary super hael Ref. [] Ot Outer (N,K,D) KD) ode Ier (,k,d) kd) ode over GF( k ) over GF() Chael Ier deoder Outer deoder For example, RS(55,39,8) ode a e omied with the (,8,3) sigle parity hek ode i the oateatio sheme (55,398,4)

33 Iterleaved Codes Two RS odes a e omied i a oateated sheme y iterleavig. A iterleaved ode is otaied y takig L odewords (of legth N) of a give ode x j =(x j,xx j,,xx jn )(j=,,,l), ad formig the ew odeword y iterleavig the L odewords as follows y i =(x,x,..,x L, x,x,..,x L,,x N,x N,..,x LN ). The proess of iterleavig a e visualized as the proess of formig a LxN matrix of L odewords writte row y row ad trasmittig the matrix olum y olum, as give elow x x x N x x x N x L x L x LN The parameter L is kow as the iterleavig degree. 33

34 Produt Codes Aother way to deal with urst errors is to arrage two RS odes ito a produt ode: k k k k iformatio symols Cheks o rows Cheks Cheks o olums o heks A produt ode (proposed y Elias ) is a (,k k,d d ) ode i whih odewords form a x array suh that eah row is a odeword from a (,k,d ) ode C, ad eah olum is a odeword from a (,k,d ) ode C ; with i, k i ad d i (i=,) eig the odeword legth, dimesio ad miimum distae, respetively, of ith ompoet ode. Both iary (suh as iary BCH odes) ad oiary odes (suh as RS odes) may e arraged i the turo produt maer. It is possile to show that the miimum distae of a produt odes is the produt of miimum distaes of ompoet odes. It is straightforwardly to show that the produt ode is ale to orret the urst error of legth =max(, ), where i is the urst error apaility of ompoet ode i=,. 34

35 BER Performae 0 - AWGN: Uoded RS(55,39); R=0.94 RS(55,3); R=0.87 RS(55,39)+RS(55,39); R=0.88 RS(55,3)+RS(55,3); R=0.76 RS(55,3)+RS(55,39); R=0.8 RS(55,39)+RS(55,3); R= Bit-erro or rate, BER Q-fator, 0 log 0 Q [db] 35

36 Turo Codes The turo odes a e osidered as the geeralizatio of the oateatio of odes i whih, h durig iterative ti deodig, di the deoders d iterhage the soft messages ertai umer of times. Oe possile implemetatio of turo eoder ad deoder ased o systemati ovolutioal or lok odes is give elow. Iput m v 0 Eoder Iterleaver m Eoder Eoder Puturig mehaism Output v, v Deiterleavig Deoder r 0 Deoder Iterleavig r r Iterleavig Deoder r r Output Deiterleavig ig 36

37 Low-Desity Parity Chek (LDPC) Codes Defiitio. A low-desity parity-hek (LDPC) ode is a liear lok ode for whih the parity-hek matrix H has a low desity of s. Defiitio. A regular (, k) LDPC ode is a liear lok ode whose parity-hek matrix H otais exatly W 's per olum ad exatly W r = W (/m) s per row, where W << m. Wr W W The ode rate r = k/ a e omputed from r W r W r W 3 is eessary for a good LDPC odes (Gallager) Defiitio. A ipartite graph (Taer graph) is a graph (odes or verties oeted y udireted edges) whose odes may e separated ito two lasses, ad where edges may oly oet two odes ot residig i the same lass. Taer graph of a ode is draw aordig to the followig rule: hek ode is oeted to variale ode v wheever elemet h v i H is a Girth: the shortest yle i Taer graph. 37

38 Example. (0, 5) lok ode with W = ad W r = W (/m) =4. Quasi-Cyli (QC)-LDPC Codes The parity hek-matrix of QC-LDPC odes a e represeted y: I I I... I I P S P S... P S S S S H I P P... P I P P... P rs rs rs 38

39 BER Performae of Large-Girth LDPC Codes Bit-erro or ratio, BER AWGN Uoded OOK RS(55,39)+RS(55,3) (R=0.8) RS(55,3) (R=0.875) RS(55,39) (R=0.937) BCH(8,3)xBCH(56,39) (R=0.8) LDPC(8547,69) (R=0.8, lattie, g=8, r=4) LDPC(46,343) (R=0.85, PG(, 6 ), g=6, r=65) LDPC(430,34), (R=0.75, CIDS, g=8, r=3) Large girth QC LDPC odes: LDPC(8540,6835) (R=0.8, g=8, r=4) LDPC(6935,3550) (R=0.8, g=0, r=3) doule 4 its 3 its LDPC(405,9) (R=0.8, g=0, r=3) Q-fator, Q [db] (per iformatio it) The girth-0 LDPC(405,9) ode of rate 0.8 (ad olum weight 3) outperforms the oateatio RS(55,39)+RS(55,3) (of rate 0.8) y 3.35 db ad RS(55,39) y 4.75 db oth at BER of

40 Referees S. Li, D. Costello, Error Cotrol Codig: Fudametals ad Appliatios, d Ed., Pretie Hall, 004. S. Hayki, Commuiatio Systems, 4th Ed., Joh Wiley & Sos, I., 00. J. B. Aderso, S. Moha, Soure ad Chael Codig: A Algorithmi Approah, Kluwer Aademi Pulishers, 99. I. B. Djordjevi, W. Rya, ad B. Vasi, Codig for Optial Chaels. Spriger, Mar